But there's only ever going to be a fixed number of bitcoin - does it all just stop working when it's all mined out?

The mining reward is not supposed to be the end state of the system. It is a subsidy, reducing over time, to get the system established.

Once established, the idea is that transaction fees are used and the distributed nature of the network causes a natural competition to be the most efficient so miners can charge the lowest fees while making a profit.

Once all bitcoins are mined, miners are supposed to fund themselves solely through transaction fees, as they partially already do today.

Iirc miners get the gas fees

I guess it's a different scenario but if all the existing miners were turned off today, there is nothing to stop me running the open source bitcoin software on a spare computer myself the day after. I guess if you ban that for all time and all places bitcoin would be over.

My impression of the intent of the statement was "disable mining altogether" not "turn off the existing physical mining rigs"

And if you turned off all the electricity in the world a lot of companies would go bankrupt. What's the point? It's not going to happen.